Vibration Of Flexible Plate Research Articles

Multi-view videogrammetry for the measurement of plate flexural vibration

This paper describes a simulation study of the measurement of the flexural deflection of a thin plate, excited by a transverse force at the fundamental frequency of the resonance modes, using an optical system consisting of multiple cameras. The estimate of the amplitude of the vibration is obtained starting from two or more sequences of images taken by the cameras, by means of triangulation. In this paper, we assume that the system is calibrated, and we focus on the error resulting from pixellisation. The simplified optical model is used to simulate how the accuracy of the measurements varies with respect to: the distance of the cameras from the structure; the opening angle between pairs of cameras; the resolution of the cameras and their number. The aim of the work is to evaluate the influence of the cameras arrangement on the measurement quality, and in particular if the use of a number greater than 2 cameras can allow a reduction of the measurement error, allowing a favorable compromise between spatial and temporal resolution of the cameras. As an example, the average error of the reconstruction of the first flexural deflection shape falls from 12% to 4%, when the measurement setup passes from 2 cameras to 12 cameras in the considered setup.

Modeling of Thin Plate Flexural Vibrations by Partition of Unity Finite Element Method

This paper presents a conforming thin plate bending element based on the Partition of Unity Finite Element Method (PUFEM) for the simulation of steady-state forced vibration. The issue of ensuring the continuity of displacement and slope between elements is addressed by the use of cubic Hermite-type Partition of Unity (PU) functions. With appropriate PU functions, the PUFEM allows the incorporation of the special enrichment functions into the finite elements to better cope with plate oscillations in a broad frequency band. The enrichment strategies consist of the sum of a power series up to a given order and a combination of progressive flexural wave solutions with polynomials. The applicability and the effectiveness of the PUFEM plate elements are first verified via the structural frequency response. Investigation is then carried out to analyze the role of polynomial enrichment orders and enriched plane wave distributions for achieving good computational performance in terms of accuracy and data reduction. Numerical results show that the PUFEM with high-order polynomials and hybrid wave-polynomial combinations can provide highly accurate prediction results by using reduced degrees of freedom and improved rate of convergence, as compared with the classical FEM.

Nonlinear tuning curve vibration using a column of non-wetted or wetted glass beads vibrating over a clamped elastic plate: Case for air-borne sound excitation

A soil plate oscillator (SPO) apparatus consists of two circular flanges sandwiching and clamping a thin circular elastic plate. The apparatus can model certain aspects of the nonlinear acoustic landmine detection problem which involves the interaction of granular material with an elastic plate in flexural vibration. Uniform spherical glass beads—representing a nonlinear mesoscopic elastic material—are supported at the bottom by the acrylic plate (4.5 inch diam, 1/8 inch thick) and stiff cylindrical sidewalls of the upper flange. Two 3 in. diam loud speakers placed above the bead column are driven with a swept sinusoidal signal applied to a constant current amplifier to generate air-borne sound excitation from 50 to 1250 Hz. A small accelerometer fastened to the underside of the plate at the center measures the response. Separate tuning curve experiments are performed using a fixed column of 350 grams using 2,3,…,10 mm diameter soda-lime glass beads—wetted or non-wetted (mineral-oil). With dry beads, backbone curves (peak acceleration vs. corresponding resonant frequency) exhibit a linear region with comparable slopes, while detailed back-bone curvature vs. bead diameter reveals more structure. Wet beads exhibit less non-linear tuning curve behavior.

Multiple low-frequency broad band gaps generated by a phononic crystal of periodic circular cavity sandwich plates

We propose a new type of phononic crystal (PnC) composed of a periodic alternation of circular cavity sandwich plates. In the low-frequency regime, the crystal can modulate the propagation of flexural waves. Governing equations are deduced basing on the classical theory of coupled extensional and flexural vibrations of plates. The dispersion relation of the infinite PnC is calculated by combining the transfer matrix method with Bloch theory. The dynamic response of the PnC with finite unit cells is further studied with finite element analysis. An experiment is carried out to demonstrate the performance of the PnC in vibration isolation. Numerical results and experimental results both illustrate that the proposed PnC can generate several wide low-frequency Bragg band gaps providing strong attenuation. The dependence of band gaps upon geometric and material parameters is also analyzed in detail in view of vibration isolation applications.

Fluid-structure-acoustic coupling for a flat plate

The present work deals with the aeroacoustic sound radiated by a forward–backward facing step in combination with a flexible wall behind the step. A numerical flow computation with coupled aeroacoustic and vibroacoustic simulation was carried out. The structural deformations of the oscillating plate like structure in the wake of the forward–backward facing step were considered to be small and therefore not affecting the flow field. The presented approach enables a separate consideration for the aeroacoustic as well as the structural borne noise. The influence of the interactions of the acoustic medium with the flexible structure on the vibroacoustic sound radiation is investigated. One-sided and two-sided coupling approaches for the vibroacoustic analysis are introduced. The two-sided vibroacoustic computation allows for considering the damping influence of the ambient fluid on the flexible plate vibration and therefore on the sound radiation. Additional to the simulations, aeroacoustic measurements in an acoustic wind tunnel were performed for validation purposes.

Directivity pattern of the sound radiated from axisymmetric stepped plates.

For the purpose of optimal design and efficient utilization of the kind of stepped plate radiator in air, in this contribution, an approach for calculation of the directivity pattern of the sound radiated from a stepped plate in flexural vibration with a free edge is developed based on Kirchhoff-Love hypothesis and Rayleigh integral principle. Experimental tests of directivity pattern for a fabricated flat plate and two fabricated plates with one and two step radiators were carried out. It shows that the configuration of the measured directivity patterns by the proposed analytic approach is similar to those of the calculated approach. Comparison of the agreement between the calculated directivity pattern of a stepped plate and its corresponding theoretical piston show that the former radiator is equivalent to the latter, and the diffraction field generated by the unbaffled upper surface may be small. It also shows that the directivity pattern of a stepped radiator is independent of the metallic material but dependent on the thickness of base plate and resonant frequency. The thicker the thickness of base plate, the more directive the radiation is. The proposed analytic approach in this work may be adopted for any other plates with multi-steps.

Experimental Study of Flexible Plate Vibration Control by Using Two-Loop Sliding Mode Control Strategy

It is well known that intelligent control theory has been used in many research fields, novel modeling method (DROMM) is used for flexible rectangular active vibration control, and then the validity of new model is confirmed by comparing finite element model with new model. In this paper, taking advantage of the dynamics of flexible rectangular plate, a two-loop sliding mode (TSM) MIMO approach is introduced for designing multiple-input multiple-output continuous vibration control system, which can overcome uncertainties, disturbances or unstable dynamics. An illustrative example is given in order to show the feasibility of the method. Numerical simulations and experiment confirm the effectiveness of the proposed TSM MIMO controller.

Solutions of dynamic and acoustic responses of a clamped rectangular plate in thermal environments

This work presents an investigation of vibration and acoustic response characters of a clamped rectangular thin plate in thermal environments. The general form of governing equation of plate flexural vibration with considering thermal loads is established, and the influence of uniform temperature change on response characters is studied in detail. A set of double sinusoidal mode shape functions for fully clamped boundaries is used to describe the displacement distribution. The governing equation is solved with Fourier series expansion to discuss the natural vibration and dynamic responses of a clamped plate in thermal environments. Accordingly, acoustic radiation characteristics are obtained based on Rayleigh integral. Results show that natural frequencies decrease with the increment of plate temperature, and the first natural frequency is much more sensitive to thermal environment changes. Response curves of plate vibration and radiated sound power shift toward lower frequency range, and the response amplitudes of the first resonant peak of the two responses present opposite variation tendencies. These phenomena are verified with finite element and boundary element simulations. Thermal loads reduce the radiation efficiency of the plate obviously below the critical frequency, but the maximum value almost remains unchanged.

Design of Multi-step Plate in Flexural Vibration

Combing the stepped thin plate in flexural vibration with the longitudinal vibration transducer into a compound of radiator, it has a numerous applications in air in high power ultrasonic field due to its extensive radiated surface and high radiated efficiency. However, design of this kind of radiator appears very complicated for a number of special functions and equations to be solved in its frequency equation. Taking the plate in flexural vibration with two steps at a free edge as an example, an algorithm for design of this kind of radiator by numerical calculation and program is proposed under given conditions of operating frequency, base thickness and material. It is shown that the tested results are in good agreement with the theoretical ones. The study provides a new method for design of the multi-step plate in flexural vibration.

Radiation impedance and equivalent circuit for piezoelectric ultrasonic composite transducers of vibrational mode-conversion

The piezoelectric ultrasonic composite transducer, which can be used in either gas or liquid media, is studied in this paper. The composite transducer is composed of a longitudinal sandwich piezoelectric transducer, a mechanical transformer, and a metal circular plate in flexural vibration. Acoustic radiation is produced by the flexural circular plate, which is excited by the longitudinal sandwich transducer and transformer. Based on the classic flexural theory of plates, the equivalent lumped parameters for a plate in axially symmetric flexural vibration with free boundary conditions are obtained. The radiation impedance of the plate is derived and the relationship between the radiation impedance and the frequency is analyzed. The equivalent circuits for the plate in flexural vibration and the composite transducer are given. The vibrational modes and the harmonic response of the composite piezoelectric transducer are simulated by the numerical method. Based on the theoretical and numerical analysis, two composite piezoelectric ultrasonic transducers are designed and manufactured, their admittance-frequency curves are measured, and the resonance frequency is obtained. The flexural vibrational displacement distribution of the transducer is measured with a laser scanning vibrometer. It is shown that the theoretical results are in good agreement with the measured resonance frequency and the displacement distribution.

Study on the high power air-coupled ultrasonic compound transducer

The high power air-coupled compound ultrasonic transducer in flexural vibration is studied. The transducer consists of a sandwich longitudinal piezoelectric transducer and a circular thin plate in flexural vibration. The resonance frequency equation and the equivalent circuit of a circular radiator with clamped boundary condition are derived. The resonance frequency equation and the equivalent circuit of the compound transducer are also obtained. The radiated acoustic field of the circular thin plate radiator is analyzed and the directional pattern is calculated. It can be seen that when the vibrational order of the circular thin plate in flexural vibration is increased, the radiated acoustic field becomes complex.

The vibration of flexible plate caused by vortices using fluid-solid coupled FEM

The vibration of flexible plate caused by vortices using fluid-solid coupled FEM

A strain energy approach to the flexural damping of layered plates

The flexural damping factor of multilayered plates is developed using a strain energy approach. The derivation is based on the exact, steady-state solution for a plate subjected to boundary conditions that are harmonic in time and position along the plate. The distribution of elastic strain energy contained within the individual layers of the plate is developed, from which an effective, flexural damping factor is determined. Comparisons to the widely used RKU analysis [D. Ross, E. E. Ungar, and E. M. Kerwin Jr., ‘‘Damping of Plate Flexural Vibrations by Means of Viscoelastic Laminae,’’ ASME, Section 3, pp. 49–87 (1959)] are made for selected cases of constrained and unconstrained layer damping. Good agreement with RKU theory is demonstrated for the selected cases. Advantages of the present analytic approach include: (1) The ability to model the effect of fluid loading; (2) Unlimited number of plate layers; and (3) The handling of compressible layers and associated loss mechanisms. Analytical estimates of damping factors are compared with measurements made on steel bars treated with an unconstrained damping layer. Stress-state assumptions are shown to be a major factor affecting theory-data agreement.

Mathematical model of dissipative parametric vibrations of flexible plates with nonhomogeneous boundary conditions

In this work, parametric vibrations of flexible squared plates with changeable boundary conditions along their contours are studied. The known T. von Kármán equations serve as a mathematical model. This continuous system is reduced to a discrete one through the method of finite approximations of O(h4) order, which is solved further by the fourth‐order Runge‐Kutta technique. New scenarios of transition from harmonic to chaotic vibrations are reported.

Flexural vibration of rectangular plates subjected to sinusoidally distributed compressive loading on two opposite sides

Natural frequencies and mode shapes of rectangular plates subjected to sinusoidally distributed in-plane compressive loading on two opposite edges were considered in this work. Although vibration analysis of rectangular plates was investigated by numerous authors, flexural vibration of plates with nonlinearly distributed in-plane loading received much less attention. However, certain problems involving thermal stresses may have such nonlinear loading conditions in the plane of the plate. The analysis procedure for the title problem involves first finding a plane elasticity solution for the in-plane problem satisfying all boundary conditions. Then using this in- plane solution, flexural vibration analysis has to be carried out. In a related work investigated recently by the present authors, an in-plane elasticity solution was developed and was compared with existing analytical solutions in the literature and with the finite element method. One aspect of this study was the stress diffusion (i.e., reduction of stress) phenomenon along the length of the plate as well as the presence of all three in-plane stress components. Using this in-plane solution, vibration analysis of rectangular plates was carried out for various plate aspect ratios and the results were compared with corresponding values from finite element analysis.

A theory for flexural vibration of thin plates with thermoelastic damping

Thermoelastic damping in thin plates can be the dominant loss mechanism under certain circumstances, e.g., as demonstrated by recent measurements for MEMS paddle oscillators. However, modeling of thermoelastic damping in flexural vibration of plates has lagged, despite the classical theory of Zener for beams. This talk shows how the thinness of the plate permits a very accurate approximation to the fully coupled system of stress and temperature. In this asymptotically valid theory, the flexural plate equations are modified by damping terms of two types. The first are viscosity-like terms in the plate equation of motion. These depend on the local curvature, being zero at saddles. The damping also modifies the boundary moment and shear force. Both the bulk and boundary effects are important in estimating the Q for resonant modes. The theory is derived from a generalized form of Hamilton’s principle, using Kirchhoff’s kinematic assumption and asymptotic approximations for small thermal coupling valid for all relevant materials. Modal damping can be estimated simply for certain boundary conditions, including fixed edges. Plates with free or partially free edges present a more interesting challenge, and various alternative estimates of Q are presented and illustrated numerically for rectangular plates and MEMS structures.

Natural frequencies and mode shapes of flexural vibration of plates: laser-interferometry detection and solutions by Ritz's method

This paper presents an experimental and theoretical study of flexural symmetric vibration modes of a linear elastic plate. A laser interferometer is used as detector of the free vibration of a rectangular parallelepiped-shaped aluminium plate. The vibration spectrum gives the lowest natural frequencies of the sample. Assumption that the vibration of the plates may be described by some approximate theories is proven to be inconsistent. The Ritz method, with products of powers of the co-ordinates as basis functions, is applied to obtain the lowest flexural natural frequencies. Three-dimensional solutions are obtained, unlike those provided by simpler theories. The experimental results are compared with the numerical predictions and a good agreement is obtained. Finally, forced motion is applied to the centre of the plate and the out-of-plane and in-plane displacement components for the first symmetric mode are measured. A good fit of the calculated values to the experimental values is found.

Mode analysis of structures using the Fourier p-element method

The Fourier p-element method is an improvement to the finite element method, and is particularly suitable for vibration analysis due to the well-behaved Fourier series. In this paper, an iteration procedure is presented for solving the resulting nonlinear eigenvalue problem. Three types of Fourier version shape functions are constructed for analyzing the circular shaft torsional vibration, the plate in-plane vibration and annular plate flexural vibration modes, respectively. The numerical results show that this method can achieve higher accuracy and converge much faster than the FEM based on polynomial interpolation, especially for higher mode analysis.

ABSOLUTE INSTABILITY OR CHAOS? — A TRIBUTE TO DR. DAVID G. CRIGHTON

The stabilities of an elastic plate clamped on an infinite, rigid baffle subject to any time dependent force excitation in the presence of mean flow are examined. The mechanisms that can cause plate flexural vibrations to be absolute unstable when the mean flow speed exceeds a critical value are revealed. Results show that the instabilities of an elastic plate are mainly caused by an added stiffness due to acoustic radiation in mean flow, but controlled by the structural nonlinearities. This added stiffness is shown to be negative and increase quadratically with the mean flow speed. Hence, as the mean flow speed approaches a critical value, the added stiffness may null the overall stiffness of the plate, leading to an unstable condition. Note that without the inclusion of the structural nonlinearities, the plate has only one equilibrium position, namely, its undeformed flat position. Under this condition, the amplitude of plate flexural vibration would grow exponentially in time everywhere, known as absolute instability. With the inclusion of structural nonlinearities, the plate may possess multiple equilibrium positions. When the mean flow speed exceeds the critical values, the plate may be unstable and jump from one equilibrium position to another. Since this jumping is random, the plate flexural vibration may seem chaotic.

STUDY ON THE RADIATION ACOUSTIC FIELD OF RECTANGULAR RADIATORS IN FLEXURAL VIBRATION

In this paper, the radiation acoustic field of a rectangular radiator in flexural vibration is studied. The radiator is a rectangular thin plate in flexural vibration with simply supported boundary conditions. Based on the theory of Rayleigh integral, the acoustic pressure distribution of the radiator in its far field is obtained analytically and the near acoustic field of the radiator is also computed numerically. In the near field, the three-dimensional acoustic pressure distribution is obtained. In the far field, the two- and three-dimensional acoustic pressure distribution patterns are calculated. The dependence of the radiation acoustic pressure of the radiator on the vibrational order of the plate in flexural vibration is analyzed. It is demonstrated that the radiation acoustic pressure along the central axis of the radiator is always zero in the near and far acoustic fields, and the radiation near acoustic field and the far acoustic field are both directional. This is different from the acoustic field generated by an oscillating piston. On the other hand, when the vibrational order is increased, the acoustic pressure distribution of the radiator becomes complex.